Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions

Everaldo S. Medeiros,Claudianor O. Alves,Paulo C. Carrião

doi:10.1155/s1085337504310018

Abstract

We study the existence and multiplicity of solutions for a class of quasilinear elliptic problem in exterior domain with Neumann boundary conditions.

Highlights

In this paper, we are concerned with the existence and multiplicity of solutions for the following class of quasilinear elliptic problem with Neumann conditions:−∆pu + |u|p−2u = Q(x) f (u) in RN \ Ω, ∂u ∂η = on ∂Ω, (1.1)where Ω ⊂ RN is a bounded domain with smooth boundary, 1 < p < N, and ∆pu is the p-Laplacian operator, that is, ∆ p u N i=1 ∂ ∂xi
We study the existence and multiplicity of solutions for a class of quasilinear elliptic problem in exterior domain with Neumann boundary conditions
We are concerned with the existence and multiplicity of solutions for the following class of quasilinear elliptic problem with Neumann conditions:

Summary

Introduction

We are concerned with the existence and multiplicity of solutions for the following class of quasilinear elliptic problem with Neumann conditions:. Using the ground-state obtained in the above theorem together with some estimates given in Sections 4 and 5, we establish a second theorem which shows the existence of a nodal solution. For this result, we will need the following hypothesis:. In the proof of Theorems 1.1 and 1.2, we used variational methods and adapted some arguments explored by Cao in [6] These results complete the study made in [6] in the sense that we consider the p-Laplacian operator and a general class of nonlinearity

Technical lemmas

Behavior of the Palais-Smale sequence

Existence of ground-state solution

Existence of nodal solution